Testing ETE model, temperature dependences of PALS data

Abstract

Three porous glasses produced by sol-gel method were investigated using Positron annihilation lifetime spectroscopy (PALS) method. The theoretical curves calculated from the pick-off annihilation model for cylindrical and spherical free volumes were compared to the results of PALS measurements done in the temperature range from 100 to 500 K. The o-Ps mean lifetime values in the pores found in the experiment change with temperature in the glasses containing pores of R < 1.5 nm in the range up to 40 ns, while the model predicts these changes less then 10 ns. The meaningful discrepancy between experimental data and model predictions was found at low temperatures. The discrepancy diminishes with increasing pore sizes. In the glass containing the largest pores, R~ 2 nm, experimental values of o-Ps lifetime agree well with model predictions. . In recent 10 years PALS began to be applied to porous materials. Such possibility was opened by, so called, extended Tao -Eldrup model (ETE). The simple version of the model mentioned above [1,2] gives the possibility to determine the free volume size from the o-Ps lifetime value, assuming a spherical void shape with a radius R. As the Ps atom is trapped in the potential well whose depth is equal to the Ps work function, for convenience of calculations it is commonly accepted to substitute that potential by infinitely deep one with the radius R 0 = R + ∆R. The ∆R = 0.17 nm is an empirically fitted parameter representing the overlap of the Ps wave function with those of the electrons of the medium. It means that inside the void (below R value) the electron density is assumed to be zero, and in the layer of ∆R thickness it is constant. The infinitely long capillary model of pores is commonly accepted in physical chemistry, so it is justified to use such a geometry of free volume in application to porous media. For the pores with a radius over 1 nm another important modification of the Tao-Eldrup model is needed. It follows from the possibility to populate the levels in the potential well located above the ground state. If thermal equilibrium is achieved, one should observe a single decay rate averaged over all populated states. It is worth mentioning that because population of the excited level is temperature dependent, the model explains lowering of o-Ps lifetimes with increasing temperature, as observed in porous media. A full discussion of the modifications introduced by the proposed model is given in Refs. [4,5].

Principles of positron porosimetry

“…Particularly, at low temperature, the (T) relation does not follow the dependence expected by the ETE model for numerous samples [42][43][44][45]. Attempts to explain this behavior as thermal expansion, adsorption of gas remnants on pore walls, or o-Ps localization [46] were taken.…”